Breedsmic temperaments: Difference between revisions

The breedsma is also the amount by which four stacked [[10/7]] intervals exceed 25/6: 10000/2401 × 2401/2400 = 10000/2400 = 25/6, which is two octaves above the classic chromatic semitone, [[25/24]]. We might note also that (49/40)(10/7) = 7/4 and (49/40)(10/7)² = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40~60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.

* ''[[Decimal]]'' (+25/24, 49/48 or 50/49) → [[Dicot family #Decimal|Dicot family]]

* ''[[Beatles]]'' (+64/63 or 686/675) → [[Archytas clan #Beatles|Archytas clan]]

* [[Myna]] (+126/125) → [[Starling temperaments #Myna|Starling temperaments]]

* ''[[Greenwood]]'' (+405/392 or 1323/1280) → [[Whitewood family #Greenwood|Whitewood family]]

* ''[[Amicable]]'' (+1600000/1594323) → [[Amity family #Amicable|Amity family]]

* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]]

Considered below are tertiaseptal, emmthird, hemififths, osiris, quasiorwell, quinmite, newt, septidiasemi, subneutral, maviloid, lockerbie, unthirds, neominor, catafourth, cotritone, fibo, quasimoha, mintone, gorgik, hemigoldis, and surmarvelpyth, in the order of increasing [[badness]].  

Aside from the breedsma, tertiaseptal tempers out [[65625/65536]], the horwell comma, [[703125/702464]], the meter, and [[2100875/2097152]], the rainy comma. It can be described as the {{nowrap| 31 & 171 }} temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo]] makes for an excellent tuning, although 171edo - [[31edo]] = [[140edo]] also makes sense, and in very high limits 140edo + 171edo = [[311edo]] is especially notable. The 15- or 16-note mos can be used to explore no-threes harmony, and the 31-note mos gives plenty of room for those as well.

==== 19-limit ====

Subgroup: 2.3.5.7.11.13.17.19

 

 

 

 

 

 

==== 23-limit ====

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 595/594, 625/624, 833/832, 875/874, 1105/1104, 1156/1155, 1216/1215

Mapping: {{mapping| 1 -19 7 0 112 43 49 -94 114 | 0 22 -5 3 -116 -42 -48 105 -117 }}

* WE: ~2 = 1200.0101{{c}}, ~44/23 = 1122.8418{{c}} (~23/22 = 77.1683{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8323{{c}} (~23/22 = 77.1677{{c}})

{{Optimal ET sequence|legend=0| 140, 311, 762g }}

Badness (Sintel): 1.08

==== 29-limit ====

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 595/594, 625/624, 784/783, 833/832, 875/874, 1015/1014, 1105/1104, 1156/1155

Mapping: {{mapping| 1 -19 7 0 112 43 49 -94 114 61 | 0 22 -5 3 -116 -42 -48 105 -117 -60 }}

* WE: ~2 = 1200.0007{{c}}, ~44/23 = 1122.8332{{c}} (~23/22 = 77.1675{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8326{{c}} (~23/22 = 77.1674{{c}})

 

 

 

Comma list: 595/594, 625/624, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014

 

 

 

Badness (Sintel): 1.18

 

 

 

Mapping: {{mapping| 1 -19 7 0 112 43 49 -94 114 61 -83 81 | 0 22 -5 3 -116 -42 -48 105 -117 -60 94 -81 }}

 

 

 

 

 

 

 

 

 

Badness (Sintel): 1.20

The generator for emmthird is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.

This has been logged as ''semihemififths'' in Graham Breed's temperament finder, but ''quadrafifths'' arguably makes more sense because it straight-up splits the fifth in four.  

In addition to 2401/2400, quasiorwell tempers out the quasiorwellisma, 29360128/29296875 ({{monzo| 22 -1 -10 1 }}). It has a generator 1024/875, which is 6144/6125 more than 7/6. It may be described as the 31 & 270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)^1/8, giving just 7's, or 384^1/38, giving pure fifths.

The generator for quinmite is quasi-tempered minor third [[25/21]], flatter than 6/5 by the starling comma, [[126/125]]. It is also generated by 1/5 of minor tenth 12/5, and its name is a play on the words "quintans" (Latin for "one fifth") and "minor tenth", given by [[Petr Pařízek]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo! Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>.

Aside from 2401/2400, [[septidiasemi]] tempers out 2152828125/2147483648 in the 7-limit. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of [[15/14]]). It is an excellent tuning for 2.3.5.7.13 and 2.3.5.7.13.17 subgroups rather than full 13- and 17-limit.

: mpping generators: ~2, ~15/14

The ''sedia'' temperament (10 & 161) is an 11-limit extension of the septidiasemi, which tempers out 243/242 and 441/440.

Lockerbie can be described as the {{nowrap| 103 & 270 }} temperament. Its generator is [[120/77]] or [[77/60]]. An obvious tuning is given by 270edo, but [[373edo]] and especially [[643edo]] work as well.  

The temperament derives its name from the {{w|Lockerbie|Scottish town}}, where a {{w|Pan Am Flight 103|flight numbered 103}} crashed with 270 casualties, and the temperament is defined as 103 & 270, hence the name. The name is proposed by Eliora, who favours it due to simplicity, ease of pronunciation and relation to numbers 103 and 270.  

Lockerbie also has a unique extension that adds the 41st harmonic such that the generator below 600 cents is also on the same step in 103 or 270 as [[41/32]], which means that [[616/615]] is tempered out.

Despite the complexity of its mapping, unthirds is an important temperament to the structure of the [[11-limit]]; this is hinted at by unthirds' representation as the [[72edo|72]] & [[311edo|311]] temperament, the [[Temperament merging|join]] of two tuning systems well-known for their high accuracy in the 11-limit and [[41-limit]] respectively. It is generated by the interval of [[14/11]] ('''un'''decimal major '''third''', hence the name) tuned less than a cent flat, and the 23-note [[MOS]] this interval generates serves as a well temperament of, of all things, [[23edo]]. The 49-note MOS is needed to access the 3rd, 5th, 7th, and 11th harmonics, however.

The commas it tempers out include the [[breedsma]] (2401/2400), the [[lehmerisma]] (3025/3024), the [[pine comma]] (4000/3993), the [[unisquary comma]] (12005/11979), the [[argyria]] (41503/41472), and 42875/42768, all of which appear individually in various 11-limit systems. It is also notable that there is a [[restriction]] of the temperament to the 2.5/3.7/3.11/3 [[fractional subgroup]] that tempers out 3025/3024 and 12005/11979, which is of considerably less complexity, and which is shared with [[sqrtphi]] (whose generator is tuned flat of 72edo's).

The generator for neominor temperament is tridecimal minor third [[13/11]], also known as ''Neo-gothic minor third''.

In addition to 2401/2400, mintone tempers out 177147/175000 ({{monzo| -3 11 -5 -1 }}) in the 7-limit; 243/242, 441/440, and 43923/43750 in the 11-limit. It has a generator tuned around 49/44. It may be described as the 58 & 103 temperament, and as one might expect, 25\161 makes for an excellent tuning choice.

: ''For the 5-limit version, see [[Diaschismic–gothmic equivalence continuum #Goldis]].''

 

Though fairly complex in the [[7-limit]], hemigoldis does a lot better in badness metrics than pure 5-limit goldis, and yet again has many possible extensions to other primes. For example, two periods minus six generators yields a "tetracot second" which can be interpreted as ~[[21/19]] to add prime 19 or perhaps more accurately ~[[31/28]] to add prime 7, or even simply as ~[[32/29]] to add prime 29, though the other two have the benefit of clearly connecting to the 7-limit representation. Note that again [[89edo]] is a possible tuning for combining it with flat nestoria and not appearing in the optimal ET sequence.

''Surmarvelpyth'' is named for the generator fifth, 675/448 being 225/224 (marvel comma) sharp of 3/2. It can be described as the 311 & 431 temperament, starting with the 7-limit to the 19-limit.